Trajectory Prediction Based on Long Short-Term Memory Network and Kalman Filter Using Hurricanes As an Example

Computational Geosciences https://doi.org/10.1007/s10596-021-10037-2

ORIGINAL PAPER

Trajectory prediction based on long short-term memory network and Kalman ﬁlter using hurricanes as an example

Wanting Qin1 · Jun Tang1 · Cong Lu1 · Songyang Lao1

Received: 3 December 2019 / Accepted: 20 January 2021 © The Author(s) 2021

Abstract Trajectory data can objectively reflect the moving law of moving objects. Therefore, trajectory prediction has high application value. Hurricanes often cause incalculable losses of life and property, trajectory prediction can be an effective means to mitigate damage caused by hurricanes. With the popularization and wide application of artificial intelligence technology, from the perspective of machine learning, this paper trains a trajectory prediction model through historical trajectory data based on a long short-term memory (LSTM) network. An improved LSTM (ILSTM) trajectory prediction algorithm that improves the prediction of the simple LSTM is proposed, and the Kalman filter is used to filter the prediction results of the improved LSTM algorithm, which is called LSTM-KF. Through simulation experiments of Atlantic hurricane data from 1851 to 2016, compared to other LSTM and ILSTM algorithms, it is found that the LSTM-KF trajectory prediction algorithm has the lowest prediction error and the best prediction effect.

Keywords LSTM · Kalman filtering · One-hot · Trajectory prediction · Hurricane

1 Introduction future position of a moving object have high application value. Trajectory prediction of a moving object has become With the continuous development of satellite navigation, a hot topic in many research fields. Hurricanes, for example, wireless communication and other technologies, mobile which is a strong and deep tropical cyclone produced in intelligent devices with positioning functions are currently the Atlantic Ocean and the eastern part of the North Pacific widely used. When people use these devices, they also Ocean and is also known as a typhoon or cyclone, are a actively or passively record a large number of historical potentially valuable application [3]. Hurricane trajectories trajectories, leading to the formation of spatiotemporal are one type of spatiotemporal trajectory. As a common trajectories [1, 2]. Spatiotemporal trajectory data can natural phenomenon, hurricanes often cause great losses accurately record the activity of moving objects over a long of life and property. On October 12, 2019, the typhoon period of time and objectively reflect the law of the activity ‘Hagibis’ landed on the Izu Peninsula in Shizuoka-ken, of moving objects. Mobile communication equipment, Japan. This typhoon killed 88 people, and 7 people went animal migration, transportation and meteorological clouds missing. More than 3900 people were affected during this are examples of moving objects in specific application typhoon. In addition, the ‘Hagibis’ typhoon caused 102.73 fields. Therefore, mining the temporal and spatial patterns billion yen of losses in Japan’s agriculture, forestry, fishery contained in the historical trajectories and predicting the and other related industries. It is very important to monitor and record the trajectory of hurricanes and provide support for the analysis and forecast of hurricanes [4]. The moving object studied in this paper is the hurri- Jun Tang cane. Trajectory prediction, as the most important method [email protected] to reduce the damage caused by a hurricane, has become a hot issue in the field of trajectory research. The com- 1 College of Systems Engineering, National University mon hurricane prediction involves climate persistence, inte- of Defense Technology, Changsha 41000, China grated forecasting and probability forecasting [5, 6]. These Comput Geosci prediction methods are complex because of involving many typhoon intensity forecast model is lower than that of the factors such as phase transition, vertical advection and the regression method, demonstrating the application prospects influence of the boundary layer. DeMaria [7] introduced sta- of BP networks in typhoon intensity forecasting. The appli- tistical methods to predict the intensity of hurricanes in the cation of machine learning methods to hurricane prediction Atlantic Ocean and the East Pacific Ocean, delaying the is still a new research field. These methods have a good abil- forecast time from three days to five days. In [8], a new ity to deal with nonlinear problems and is especially suitable spatiotemporal multivariate function model was proposed for solving nonlinear problems with complex internal mech- to improve prediction. This model regarded the dimensions, anisms. Therefore, the machine learning methods can be accuracy and wind speed of hurricanes as time functions to applied in meteorology. understand the spatial and temporal trends of the path and LSTM as a neural network has been widely and used intensity of a hurricane. in text sequences or images for a long time. In this paper, Machine learning is a part of artificial intelligence. In an LSTM-KF trajectory prediction algorithm is proposed essence, machine learning enables computers to simulate by combining LSTM and Kalman filter. The simulation human learning behaviour, automatically acquire knowl- results show that the LSTM-KF algorithm has a good effect. edge and skills through learning, continuously improve The rest of the paper is organized as follows: Section 2 performance and realize artificial intelligence [9]. The com- introduces the correlation method of trajectory prediction mon algorithms of machine learning include regression and the basic mechanism of the neural network. The relevant algorithm, algorithm, support vector machine (SVM), clus- methods and technologies used in trajectory prediction are tering algorithm, dimension reduction algorithm and neural described in Section 3. In Section 4, the improved LSTM network [10]. Neural networks are currently the most influ- trajectory prediction algorithm and a trajectory prediction ential algorithms in machine learning. With the popular- algorithm based on LSTM and Kalman filter (LSTM-KF) ization and application of new meteorological observation are proposed. Finally, Atlantic hurricane data are used in equipment, it is possible to obtain a large amount of ground a simulation experiment to verify the rationality of the and upper-air data from meteorological observation stations, improved LSTM and the LSTM-KF trajectory prediction as well as satellite and radar detection data. These data algorithm in Section 5. are in different formats, such as numbers, text and images, and have spatial and temporal characteristics [11]. Machine learning methods can provide new information regarding 2 Related work how to deeply excavate the physical mechanism of weather changes hidden in these massive data, explore signals that 2.1 Trajectory prediction method indicate the evolution law of weather and establish a new weather forecasting method. In recent years, it has become At present, the active space of moving objects can be a hot spot in the fields of meteorology, mathematics or divided into restricted movement [17, 18, 31] and free computer to study the effective weather forecasting meth- movement [19–22]. Current trajectory prediction methods ods combined with machine learning. The application of mainly focus on restricted movement trajectory prediction machine learning to hurricane prediction has also achieved with certain constraints (such as road networks). Because some fruitful achievements. In [12], the method of fuzzy this kind of trajectory follows specific behaviour habits or c-means clustering was used to classify typhoon trajecto- motion patterns, useful patterns are easily found and the ries and forecast them, which is suitable for moving objects prediction results are satisfactory regardless of the accuracy with fuzzy trajectory boundaries. Song [13] combined big or effectiveness. However, in nature, free movement is more data and data mining technology, trained a long short-term frequent than limited movement and is more important to memory (LSTM) neural network, and established a typhoon predict, especially for disaster prevention and mitigation. path prediction model based on machine learning. In [14], The sampling interval of free movement trajectory data are a sparse recurrent neural network (RNN) combined with a sparse, and it is difficult to accurately capture the moving flexible topological structure was proposed to predict the direction, turning position and other moving characteristics trajectory of Atlantic hurricanes. The dynamic time warping of moving objects, which makes the prediction of the (DTW) distance between the direction of target hurricanes sparse trajectory in free space difficult in trajectory data and other hurricanes in the dataset was determined and com- research. Hurricane trajectory prediction is a kind of free pared to predict their future trajectories. Baik [15, 16]useda movement trajectory prediction. According to the prediction backpropagation (BP) network to forecast typhoon intensity cycle, trajectory prediction can be divided into short- and compared it with results from the regression method. term trajectory prediction [23] and long-term trajectory The results showed that the error of the BP network’s prediction [24]. Comput Geosci

Trajectory prediction methods can be divided into tradi- prediction. Related technologies used for moving object tional prediction methods based on a construction motion trajectory prediction are listed in Table 1. model [25–28], methods based on frequent pattern mining technology [29–31] and methods based on a machine 2.2 Overview of neural network learning [32–36]. The prediction methods based on the construction of a motion model mainly consider the cur- Neural networks (also known as artificial neural networks rent movement of moving objects, such as their speed and or ANNs) are among most powerful machine learning algo- direction, and matches the predicted movement with the rithms. The general idea of a neural network is to adopt constructed motion function for prediction. Wolfson [25] learning and training. First, the input is calculated by pro- et al. proposed a spatiotemporal model of moving objects cessing the neurons in the hidden layer and the output layer, (MOST) that took location as a dynamic attribute and then which are compared with the expected output. Then, a back- predicted the location of a mobile object based on lin- propagation learning algorithm is used to repeatedly correct ear constraints. Junghans et al. [26] proposed a model and the connection weight coefficient from the input layer to prediction method of a moving area that used the mini- the hidden layer and from the hidden layer to the output mum closed box to model the moving area and used linear layer. When the error between the output and the expected regression (LR) and recursive motion (RMF) functions to output of the neural network reaches the preset error conver- determine the evolution mode of the area. gence standard, the training is stopped to obtain a network Trajectory prediction based on frequent pattern min- model with an improved generalization ability [37]. ing technology achieves the goal of trajectory prediction Figure 1 shows the general structure of a neural network by analysing historical data in a moving object database, that consists of an input layer, hidden layer and output processing the historical data and mining the matching layer. The input layer is responsible for receiving the signal, pattern. Monreale [29] comprehensively considered the the hidden layer is responsible for the decomposition and spatiotemporal information of moving objects to build a processing of data, and the final result is integrated into the T-pattern tree and mined the motion law of the mov- output layer. The circles in each layer in Fig. 1 represent ing objects. Long [30] mined frequent paths based on an processing units that can be thought of as simulating a FP growth algorithm that introduced the speed of mov- neuron. Neurons are connected by nerve wires, which can be ing objects into the prediction and proposed a simple and assigned different weights. The core task of neural network effective trajectory prediction algorithm (E3TP). Kim et al. is to train a set of nearly perfect weights. Several processing [31] combined the characteristics of road networks, cal- neuron make up a layer, and several layers make up a culated the similarity between trajectories to search for network, which is known as a neural network. When there is candidate trajectories, evaluated the historical trajectories more than one hidden layer of the neural network is called a stored in a historical mobile database to determine the sub- deep neural network. trajectories with similar trajectories, and finally, predicted The BP neural network was the first neural network a future path by analysing the direction of the candidate described. Werbos [38] first proposed the BP neural network trajectories. in 1974, and Rumelhart et al. [39] developed the theory Machine learning methods mainly aim at mining the of the network. The learning process of a BP neural behaviour characteristics of moving objects in a historical network consists of the forward propagation of a signal trajectory and perform trajectory prediction by improving and the backpropagation of error. The cell structure of a = the Markov model, probability graph model, support vector BP neural network is shown in Fig. 2.InFig.2, Sj n + = · machine model and neural network model. Based on the i=1 wij xi bj and yj f(Sj ). f() represents the hidden Markov model, Qiao [32] proposed an adaptive transfer function, Wij represents weight, bj represents bias. parameter selection algorithm to improve the adaptability The weight and bias are acquired by the neural network of trajectory prediction in a big data environment. Li [33] through constant training. In forward propagation, yj is the introduced the concept of fuzzy trajectory to solve the output of neuron. If the actual output does not match the sharp boundary problem caused by fixed mesh generation expected output, then the error backpropagation stage is and improved the traditional LSTM model to make full entered. The purpose of backpropagation process is to adjust use of the proximity and periodic characteristics of the weights and bias to reduce the error between the output historical trajectory to improve the prediction accuracy value and the expected value. The backpropagation process of the trajectory position. Leege [34] proposed a path is summarized in references [38, 39]. When there is more prediction method based on machine learning that combined than one hidden layer of the neural network is called a and sorted the traffic flow of fixed arrival routes of airplanes deep neural network. At present, many advances in image by their actual flight path and meteorological data and recognition and speech recognition technology are derived trained the model by using historical data to perform time from the development of deep neural network. Comput Geosci

Table 1 Comparison of trajectory prediction methods

Method Restricted movement Free movement Tectonic movement model Frequent pattern mining Machine learning

Wolfson [25] Junghans [26] Prevost [27] Jeung [28] Monreale [29] Long [30] Kim [31] Qiao [32] Li [33] Leege [34]

3 Trajectory deﬁnition and long short-term as an example, Im can represent the maximum sustained memory wind speed and status of system.

Trajectory data store large amounts of position information Definition 2 Subtrajectory. A subtrajectory refers to the of moving objects at different times. A collection of data in ordered set of trajectory points within a certain trajectory, as chronological order is called a trajectory Trj. showninEq.2.

Definition 1 Trajectory. Moving objects move in geospatial Seq = {(Pm,Im) , (Pm+1,Im+1) , ··· , space, and their positions change with time. The ordered (Pm+k−1,Im+k−1)} (2) sequence of m discrete position points and related auxiliary information is defined as trajectories. Where k is the length of subtrajectory.

Trj = {(P1,I1) , (P2,I2) , ··· , (Pm,Im)} (1) Definition 3 Grid trajectory. The geographic space of a where Pm represents the spatial position information of the moving object is divided into different space areas by fixed m-th trajectory point and Im represents the relevant infor- grids (such as squares, triangles or hexagons), and a single mation of the m-th trajectory point. Taking hurricane data coordinate point of the trajectory is mapped to the space

Fig. 1 General structure of a z neural network 1 1 x 1 y 1 1 1

zk x i i k j y j

z p

xn n y p q q

Input Layer Hidden Layer Output Layer Comput Geosci

Fig. 2 Cell structure of BP x neural network 1 w bj 1 j x 2 S j y w f j 2 j

j wnj xn

area. The representation of the grid trajectory of the original long-term memory at last moment Ct−1 can be retained until trajectory Trj after gridding is shown in Eq. 3: the present moment, as shown in Eq. 4. ft = σ Wf · ht−1,xt + bf (4) G Trj = {(S1,I1) , (S2,I2) , ··· , (Sm,Im)} (3) The value of ft in each dimension is in the range of (0, 1). The information will be forgotten when f is close to 0, where Sm represents the spatial position information of t the m-th trajectory point mapped to the corresponding and the information will be retained when ft is close to 1. grid index on the grid space. As shown in Fig. 3,the Where Wf represents weight matrix, and bf represents bias. • represents the matrix product. σ represents a Sigmoid corresponding grid trajectory is Trj ={(P1,I1), (P2,I2), function, as shown in Eq. 5. ··· ,(P5,I5)} is G Trj={(S1,I1), (S2,I2), ··· ,(S5,I5)}. 1 σ(x) = (5) The long short-term memory (LSTM) network [40] 1 − e−x is a kind of time recurrent neural network (RNN) that The input gate is responsible for processing the current is specially designed to solve the long-term dependence input and updates new information selectively. The input problem of the general RNN. LSTM is widely used in gate has two steps. In Eq. 6, it can be used to control how temporal sequences now. Trajectory data is also a kind of much of the current input can be stored. The value of it temporal data, so LSTM can be applied to the trajectory [13, in each dimension is in the range of (0, 1). In Eq. 7,the 33]. The cell structure of LSTM is shown in Fig. 4.The tanh function generates a new candidate vector Ct .The cell generally includes a forget gate, an input gate and an Ct represents cell state of short-term memory at current output gate. And it is like a memory machine that constantly moment. forgets some information, remembers some information, = · + and takes into account previous input before output. In the it σ Wi ht−1,xt bi (6) cell structure of LSTM, xt is the input at current moment, ˜ ht−1 is the output at last movement, and Ct generally Ct = tanh WC · ht−1,xt + bC (7) referred to as the cell state, which is a cell state of long-term where W ,W represent weight matrix and b ,b represent memory. i c i c bias. The tanh function is as shown in Eq. 8. The forget gate, as the name implies, controls whether ex − e−x information should be forgotten. In LSTM, the forget gate tanh(x) = (8) controls how much information about the cell state of ex + e−x

Fig. 3 Gridding of trajectory sequence P S 5 5 P 4 S 4 P 2 P S S 3 2 3 S P 1 1 Comput Geosci

Fig. 4 Cell structure of LSTM

Before output gate, the cell state of long-term memory degree. Because of the spherical nature of the earth, the needs to be updated, as shown in Eq. 9. The cell state of area of each grid block is not uniform across square miles, short-term memory at current movement Ct and cell state of but since most points surround the earth’s equator, the size long-term memory at last movement Ct−1 are form new cell difference between each grid block is negligible. The grid state. trajectory, as shown in Fig. 3, is taken as the input of LSTM. ˜ The encoding of trajectory data can be divided into three Ct = ft ∗ Ct−1 + it ∗ Ct (9) steps: where ∗ represents the elementwise product. The output gate is used to selectively output information Step 1: One-hot encoding for categorical data. The grid index Sm in the grid trajectory can be one-hot from the cell. As shown in Eqs. 10 and 11, ot is a probability encoded. The status of system of hurricane data that controls how much of Ct can be used as the current is also a kind of categorical data, which can also output. The value of ot in each dimension is in the range of be one-hot encoded. The status of system has (0, 1). ht−1 is the output at time t and it is hidden state. nine types, which can also be one-hot encoded, = · + ot σ Wo ht−1,xt bo (10) such as tropical cyclone of tropical depression intensity (TD), tropical cyclone of tropical storm ht = ot ∗ tanh(Ct ) (11) where Wo is weight matrix and bo is bias. Wf ,Wi,Wt ,Wo Table 2 Nnotations in Section 3 and bf ,bi,bt ,bo represents respectively weights and bias Notation Description of different gate. These two parameters are obtained by continuous training of LSTM network using historical Trj Trajectory training data. The notations referred to in Section 3 are listed Seq Subtrajectory in Table 2. G Trj Grid trajectory σ Sigmoid function tanh tanh function

4 Proposed LSTM-KF Wf ,Wi ,Wt ,Wo Weight matrix bf ,bi ,bt ,bo Bias 4.1 Encoding of trajectory data ft ,it ,ot Output of the forget gate, input gate and output gate, between 0 and 1 Hurricane trajectory data should be converted into data that • matrix product can be received by the LSTM network through trajectory ∗ elementwise product data encoding. In this paper, the latitude and longitude Ct Cell state of long-term memory coordinates, maximum sustained wind speed and status Ct Cell state of short-term memory of system of hurricane are selected. When the hurricane ht Hidden state output at time t trajectory is meshed, the size of the mesh blocks is 1 × 1 Comput Geosci

intensity(TS), tropical cyclone of hurricane inten- The LSTM network can learn historical knowledge, but sity(HU), extratropical cyclone(EX), subtropical it may not be suitable for learning new knowledge. For cyclone of subtropical depression intensity(SD), example, the area where the hurricane trajectory appears subtropical cyclone of subtropical storm inten- may not have occurred in history or occurred only a few sity(SS), a low that is neither a tropical cyclone times, so the LSTM network may produce a large error (LO), tropical wave(WV) and disturbance(DB). between the predicted position and the real position due One-hot encoding is the most commonly used and to insufficient learning. At the same time, there are few basic coding method to convert tags into vectors. feature data available for hurricane data, and the Atlantic One-hot encoding first requires categorical values hurricane data used in this paper only include coordinates, to be mapped to integer values. Each integer value maximum sustained wind speed, status of system, so a lack is then represented as a binary vector with zero of features is also an important factor that restricts neural values except for the index position, which has network learning. a value of 1. One-hot encoding diagram of grid On average, hurricanes move at speeds ranging from index is shown in Fig. 5a and one-hot encoding 15 km/h to 20 km/h and can be as fast as 30 km/h diagram of status of system is shown in Fig. 5b. according to hurricane trajectory analysis [41, 42]. With Step 2: The numerical data are standardized so that data intervals of 6 h, the distance a hurricane can travel the range of numerical data is between [0 1]. is limited. The result of the simple LSTM trajectory To eliminate the dimensional impact between prediction algorithm selects the grid corresponding to the indicators, data standardization is needed to solve highest probability value as the prediction position. The the comparability between data indicators. After prediction of a single position may be take extreme case the original data are standardized, each index is as the prediction result, that is, the predicted grid position on the same order of magnitude, which is suitable significantly deviates from the normal position range. If for a comprehensive comparative evaluation. The the hurricane position predicted by the simple LSTM numerical data in Im are standardized by Eq. 12. trajectory prediction algorithm exceeds a certain limit, then The maximum sustained wind speed of hurricane it needs to be corrected. In this case, we improve the data is numerical data, so it can be normalized simple LSTM trajectory prediction algorithm. We select a between [0,1]. reasonable position from the candidate set by predicting multiple possible positions and filter out positions with a z − min (z)) high probability of being extreme positions. The ILSTM z = (12) max (z) − min (z) trajectory prediction algorithm is an improvement of the prediction module of the simple LSTM algorithm. And it Step 3: Connect the categorical data and numerical data is consistent with the simple LSTM algorithm in regard to to form a complete trajectory vector. The transfor- the data preprocessing module and model training module. mation from grid trajectory to a trajectory vector First, the following definitions are given: is realized by one-hot encoding and standardization. The trajectory data coding process is shown Definition 4 Subtrajectory marker point seq flag.The in Fig. 6. last point in subtrajectory Seq ={(Pm,Im) ,(Pm+1, Im+1), ··· , (Pm+k−1,Im+k−1)} is taken as the mark point 4.2 Improved LSTM (ILSTM) of the subtrajectory, that is, seq flag=(Pm+k−1,Im+k−1).

The simple LSTM trajectory prediction algorithm selects Definition 5 Grid centre coordinate Psi central, represents the subtrajectory with a sequence length of K. Then, the the coordinate corresponding to the centre point of grid Si, subtrajectories are meshed and encoded into the LSTM that is, lonsi central,latsi central . network to predict the index of (K + 1)-th grid. The flow diagram of the simple LSTM trajectory prediction Multiple grids are selected as a candidate set, and the algorithm is shown in Fig. 7. The simple LSTM trajectory order of the candidate set is sorted according to probability prediction algorithm is divided into three modules: data predicted by the simple LSTM. That is, the grid correspond- preprocessing, model training and prediction. The data ing to the first top N probability value is selected as the preprocessing modules involves gridding and coding the candidate set C = S1,S2, ··· ,Stop N of the prediction trajectory data to produce an acceptable trajectory vector results. The first grid whose Euclidean distance (13) from for LSTM. The training module is to finds the best LSTM the centre coordinates to the coordinates of seq flagis less prediction model. The prediction module predicts the next than the threshold τ is selected as predicted grid from candi- grid index according to the trained prediction model. date set C. Then, the centre coordinates of the grid selected Comput Geosci

Fig. 5 One-hot encoding S diagram of a grid trajectory 1 0 0 0 0 1 S 5 S 0 1 0 0 0 2 One-hot S encoding S 4 0 0 1 0 0 3 S S S 2 3 0 0 0 1 0 4 S 1 S 0 0 0 0 1 5

(a) One-hot encoding diagram of grid index

TU 1 0 0 0 0 0 0 0 0

TS 0 1 0 0 0 0 0 0 0

HU 0 0 1 0 0 0 0 0 0

EX 0 0 0 1 0 0 0 0 0

SD 0 0 0 0 1 0 0 0 0

SS 0 0 0 0 0 1 0 0 0

LO 0 0 0 0 0 0 1 0 0

WV 0 0 0 0 0 0 0 1 0

WV 0 0 0 0 0 0 0 0 1

(b) One-hot encoding diagram of status of system

Fig. 6 Schematic diagram of the trajectory data encoding process

Fig. 7 Flow diagram of the simple LSTM trajectory prediction algorithm Seq VSeq Grid1 1 _ 1

Seq VSeq Grid2 2 _ 2

Trajectories

Seqn VSeq_ n Gridn Trajectory Gridding and Sequence Encoded LSTM Network Grid Data preprocessing Model training prediction Comput Geosci from the candidate set are the predicted coordinates. If of the next point of each subtrajectory are predicted. If there is no grid located less than threshold τ in candidate step = 1, then n − k subtrajectory sequences will be set C, then the grid centre coordinates of seq flag are obtained, so n − k predicted coordinates will also be taken as the predicted coordinates in this paper. That is, generated. These sequences generated by the predicted lonseq flag central,latseq flag central are used as the coor- coordinates are called predicted trajectory sequences Z = dinates of the predicted trajectory. The selection of τ is {(lon1,lat1) , (lon2,lat2) , ··· , (lonn−k,latn−k)}. based on experience. The ILSTM trajectory prediction algorithm modifies the position of predicted extreme cases in In this paper, it is assumed that the generated predicted a few cases. In most cases, the first grid in the candidate trajectory sequence Z by the ILSTM method is a kind of set satisfies the conditions. The improved LSTM trajectory observation data. Then, the Kalman filter will filter the prediction algorithm is shown in Fig. 8. predicted trajectory sequence to produce a more accurate and optimal estimation.

= − 2 + − 2 dist lonsi central lonseq flag latsi central latseq flag Definition 7 Trajectory marker point Trj flag. The sub- (13) trajectory marker point seq flag of the first subtrajectory sequence extracted from each trajectory is taken as the 4.3 Trajectory prediction based on LSTM trajectory marker point Trj flagof the whole trajectory. and Kalman ﬁlter The Kalman filter is used to filter the predicted trajectory The Kalman filter [43], as a data processing technology sequence. First, the state vector of the hurricane movement that removes noise and restores real data, has been applied is defined. In this paper, the state vector of a hurricane, as in fields such as communication, navigation, guidance showninEq.14,isused. and control. Kalman filtering estimates the state of a T X(m) =[lonm,latm,Vlon,Vlat] (14) system optimally through the observation of the input and output. Because the observation data include the influence where Vlon,Vlat represent velocity components in longitude of system noise and external interference, the optimal and latitude, respectively. In hurricane trajectory data, there estimation can also be regarded as a filtering process. is no hurricane velocity information at the current moment, so for convenience we set the initial moment Vlon,Vlat Definition 6 Prediction trajectory sequence Z.Fora to 0. In this paper, when the Kalman filter is used, the trajectory Trj with length n, step is used as the system state vector of the trajectory mark point Trj flag extraction interval of the subtrajectory and k is the is input into the system as the initial value, that is, X(0) = T extraction length of the subtrajectory, and the coordinates [lonTrj flag,latTrj flag, 0, 0] .

Fig. 8 ILSTM trajectory prediction algorithmv Comput Geosci

Kalman filtering requires a discrete control process until the end of the system process, we need to update the system. The system state equation (15) and observation covariance of X(m|m) at time m. equation (16) of hurricane movement are as follows: ρ(m|m) = (I − KH)ρ(m|m − 1) (21) + = + X (m 1) AX (m) W (m) (15) So far, the entire Kalman filtering process of trajectory prediction has been completed. The function of the Kalman = + Z (m) HX(m) V (m) (16) filter in this paper is to estimate the optimal trajectory where X(m) represents the system state vector and coordinates predicted by ILSTM algorithm. The notations describes the state vector of the hurricane at time m referred to in Section 4 are listed in Table 3. (14). A represents the state transition matrix, which is used to describe the motion state transition mode from the previous time to the current time.W(m) represents the 5 Experimental results and discussion system noise, whose statistical characteristics are similar to T 5.1 Description of Data set white noise or Gaussian noise. Z(m) =[lonm,latm] is the observation vector.In this paper, the prediction trajectory sequence Z is used to represent the observation sequence, Atlantic hurricane data released by the National Hurricane which represents m-th predicted value. H is the observation Centre of the United States (www.nhc.noaa.gov/data) is matrix, and V(m)is the observation noise. used in the experiment. The dataset includes the longitude The core of the Kalman filter algorithm uses a recursive and latitude, maximum sustained wind speed, minimum algorithm to achieve the optimal state estimation model central pressure, time and other information of hurricanes. and update the current state variables by using the previous The sampling interval of this data set is 6 h. This experment estimated value and the current observed value. The state mainly deals with the latitude and longitude, maximum equation of the system is used to estimate the state of the sustained wind speed and the status of the system. The system at the next time point. If the current time of system is 165 years of hurricane data from 1851 to 2015 are selected m, then it can be estimated to the current state based on the as training sets to train the LSTM prediction model. The previous state according to the state equation of the system. hurricane data from 1851 to 2015 include 1780 hurricane trajectories and 49139 sampling points.First, the dataset x(m|m − 1) = AX(m − 1) + W(m) (17) is preprocessed, and the missing data are filled with the linear interpolation method. The trajectory map of where X(m|m − 1) is the previous state estimation and Atlantic hurricanes from 1851 to 2015 is shown in Fig. 9 X(m − 1|m − 1) is the previous state optimal estimation. (East longitude is represented by positive values and west The system result has been updated, but the covariance longitude is represented by negative values). corresponding to X(m|m − 1) has not been updated. Equation 18 represents the covariance. 5.2 Prediction error comparison experiment ρ(m|m − 1) = Aρ(m − 1|m − 1)AT + Q (18) { } The real trajectories are T = Trj1,Trj2, ··· ,Trjn .The where ρ(m|m − 1) is the covariance of X(m|m − 1). ρ(m− = ··· predicted trajectories are T Trj1,Trj2, ,Trjn . 1|m − 1) is the covariance of X(m − 1|m − 1). AT is the transpose matrix of A. Table 3 Notations in Section 4 The present state X(m|m − 1) has been estimated. Then, we obtain the measured values of the current state, that Notation Description is, the trajectory coordinates Z(m) predicted by ILSTM algorithm. Combining the estimated value and the measured seq flag Subtrajectory marker point P Grid centre coordinate value, we can obtain the optimal estimated value X(m|m) at si central time m, as shown in Eq. 19. τ Distance threshold Z Prediction trajectory sequence X(m|m) = X(m|m − 1) + K(Z(m)− HX(m|m − 1)) (19) Trj flag Trajectory marker point X(m) State vector of hurricane movement where K is the Kalman gain matrix that is shown in Eq. 20. K Kalman gain matrix In this paper, K is a 4 × 2matrix. A State transition matrix − K = ρ(m|m − 1)H T (Hρ(m|m − 1)H T + R) 1 (20) H Observation matrix Q System noise covariance matrix Until now, we produced the best estimate value X(m|m) R Observation noise covariance matrix at time m. However, to keep the Kalman filter running Comput Geosci

Fig. 9 Atlantic hurricane trajectories from 1851 to 2015

The geometric space error between the predicted trajectory The structure of LSTM network shown in Figs. 10aand point and the actual trajectory point is used as the prediction 11a is 128 * 128 * 256, the structure of LSTM network error. As shown in Eq. 22, the root mean square error shown in Figs. 10band11b is 128 * 256 * 512, the structure (RMSE ) unit is degrees. of LSTM network shown in Figs. 10cand11c is 256 * 256, and the structure of LSTM network shown in Figs. 10d n m − 2 + − 2 and 11d is 256 * 256. It can be seen from Figs. 10 and 11 1 k=1 (lonik lonik) (latik latik) RMSE = that the RMSE of the simple LSTM trajectory prediction n m i=1 algorithm is initially large. The LSTM network does not (22) learn enough at the initial moment, and the prediction error of simple LSTM algorithm is initially large. However, the Here, n represents the number of predicted trajectories RMSE of the ILSTM and LSTM-KF trajectory prediction and m represents the number of predicted trajectory points algorithms are significantly lower than those of the simple contained in each predicted trajectory. LSTM. With an increase in the LSTM network training In this experiment, hurricanes in 2016 are used as the epochs, the RMSE of the simple LSTM algorithm gradually verification set. The prediction errors of the simple LSTM decreases and becomes stable. However, the prediction algorithm, ILSTM algorithm and LSTM-KF algorithm are error of the simple LSTM is also significantly higher than compared. The setting of experimental parameters is shown that of the ILSTM and LSTM-KF trajectory prediction in Table 4. Figures 10 and 11 show the change of RSME algorithms. value with the neural network training epochs. Figure 10 To better display the RMSE of the three methods, the indicates that Q is [1,0,0,0; 0,1,0,0; 0,0,1,0; 0,0,0,1], RMSEs of epoch 71 to epoch 80 shown in Figs. 10band and Fig. 11 indicates that Q is [2,0,0,0; 0,2,0,0; 0,0,2,0; 11b are listed in Table 5. The system noise covariance 0,0,0,2]. matrix Q in Figs. 10band11b are [1,0,0,0; 0,1,0,0; 0,0,1,0; 0,0,0,1] and [2,0,0,0; 0,2,0,0; 0,0,2,0; 0,0,0,2], respectively. Table 4 Parameter settings of the prediction error comparison experiment The Network structure of LSTM in Figs. 10band11bare both 128*256*512. The observation matrix H in Figs. 10b Parameter Value and 11b are both [1,0,0,0; 0,1,0,0]. The observation noise Network structure of LSTM 128*128*256; 128*256*512; covariance matrix R in Figs. 10band11b are both [0.3, 256*256; 256*256*256 0, 0, 0.3]. It can be seen from Table 5 that the RMSE of Distance threshold τ 6 each epoch of LSTM-KF is smaller than that of ILSTM top N 20 and simple LSTM. In other words, the prediction effect State transition matrix A [1,0,1,0; 0,1,0,1; 0,0,1,0; 0,0,0,1] of LSTM-KF should be the best of the three methods. In Fig. 10b, the average prediction error of the simple LSTM Observation matrix H [1,0,0,0; 0,1,0,0] ◦ System noise covariance [1,0,0,0; 0,1,0,0; 0,0,1,0; 0,0,0,1] from epoch 71 to epoch 80 is 2.1065 , that of the ILSTM is 1.4690◦, and that of the LSTM-KF is 1.4605◦.InFig.11b, matrix Q or [2,0,0,0; 0,2,0,0; 0,0,2,0; 0,0,0,2] the average prediction error of the simple LSTM from epoch Observation noise covariance [0.3,0; 0, 0.3] 71 to epoch 80 is 2.3865◦, that of the ILSTM is 1.5141◦,and matrix R that of the LSTM-KF is 1.5017◦. Therefore, the prediction Comput Geosci

Fig. 10 Change of the RMSE with LSTM network training epochs when Q = [1,0,0,0; 0,1,0,0; 0,0,1,0; 0,0,0,1] error of LSTM-KF is the smallest of the three methods. The 5.3 Comparison experiment of single hurricane prediction error of the simple LSTM trajectory prediction trajectory prediction algorithm is approximately 2◦, and its effect is the worst of the three methods. However, the prediction error of LSTM- In this experiment, five hurricanes named ‘KARL’, KF is slightly smaller than that of ILSTM, which shows that ‘MATTHEW’, ‘NICOLE’, ‘BONNIE’ and ‘ALEX’ are the trajectory coordinates predicted by LSTM-KF are more selected from hurricanes that occurred in 2016. The simple precise than those predicted by ILSTM. It can be concluded LSTM, ILSTM and LSTM-KF algorithms are used to pre- that LSTM-KF has the best prediction effect of the three dict the trajectories of these hurricanes. The experimental methods. parameters are set as shown in Table 6.

Fig. 11 Change of the RMSE with LSTM network training epochs when Q = [2,0,0,0; 0,2,0,0; 0,0,2,0; 0,0,0,2] Comput Geosci

Table 5 RMSEsofepoch71toepoch80inFigs.10band11b (4 decimal places are reserved)

Epochs Fig. 10bFig.11b

LSTM ILSTM LSTM-KF LSTM ILSTM LSTM-KF

71 1.9795 1.4897 1.4854 2.7606 1.5401 1.5268 72 2.1406 1.4681 1.4584 2.0849 1.4859 1.4807 73 2.2656 1.4557 1.4515 2.2552 1.5472 1.5331 74 2.3020 1.4398 1.4359 3.0540 1.4551 1.4370 75 2.1322 1.4748 1.4669 2.9492 1.5269 1.5167 76 1.8742 1.4817 1.4717 2.1975 1.4902 1.4782 77 2.1561 1.4827 1.4731 2.1008 1.5131 1.4955 78 2.2247 1.4572 1.4523 2.1516 1.4858 1.4751 79 1.9575 1.4850 1.4772 2.2730 1.5291 1.5167 80 2.0322 1.4554 1.4373 2.0382 1.5680 1.5569 Averaging 2.1065 1.4690 1.4610 2.3865 1.5141 1.5017

First, the simple LSTM, ILSTM and LSTM-KF tra- 16, some predicted points of the simple LSTM algorithm jectory prediction algorithms are used to predict hurri- deviate significantly from the original trajectory. Figure 15 canes KARL and MATTHEW. The results are compared in shows the last two predicted points of the elliptical cir- Figs. 12 and 13 respectively (the trajectory sequence used cle in hurricane NICOLE, and Fig. 16 shows the predicted for training is not shown in the figures.). In Figs. 12 and points of the elliptical circle in hurricane BONNIE. These 13, the blue dotted line represents the original trajectory, the predicted trajectories deviated significantly from the orig- black dotted line represents the trajectory predicted by the inal hurricane orbit. In Fig. 16, the parts marked by the simple LSTM algorithm, the green dotted line represents boxes ‘Contrast 1’ and ‘Contrast 2’ indicate that the pre- the trajectory predicted by the ILSTM algorithm, and the dicted trajectory points of the simple LSTM algorithm show red line represents the trajectory predicted by the LSTM- obvious confusion, while the predicted trajectory points of KF algorithm. From Figs. 12 and 13, the simple LSTM, ILSTM and LSTM-KF basically follow the original trajec- ILSTM and LSTM-KF all show good prediction results, tory. Therefore, there are some defects in the simple LSTM almost according to the change trend of the original tra- trajectory prediction. The prediction accuracy of the sim- jectory, which shows that according to the learning of the ple LSTM trajectory prediction algorithm is greatly reduced history track, the neural network still has great advantages. without adequate learning. The simple LSTM trajectory While LSTM is effective, ILSTM and LSTM-KF produce prediction algorithm has a good advantage for learning better prediction results. historical knowledge, but for time-series data with fewer Then, the simple LSTM, ILSTM and LSTM-KF trajec- sample data and high mutation, the simple LSTM algo- tory prediction algorithms are used to predict hurricanes rithm easily produces a large range of prediction errors. The NICOLE, BONNIE and ALEX. The results are shown in ILSTM and LSTM-KF algorithm, which modify the simple Figs. 14, 15 and 16, respectively. From Figs. 14, 15 and LSTM algorithm, show better robustness and can reduce the errors of the simple LSTM algorithm. It can be seen from Table 6 Parameter settings of comparison experiment of single Table 5 that ILSTM performs a rough prediction of trajec- hurricane trajectory prediction tory coordinates, while LSTM-KF performs a more precise prediction of trajectory coordinates. Parameter Value

Network structure of LSTM 256*256*256 5.4 Parameter sensitivity experiment and time Distance threshold τ 6 performance analysis top N 20 State transition matrix A [1,0,1,0; 0,1,0,1; 0,0,1,0; 0,0,0,1] In order to verify the influence of different system noise Q Observation matrix H [1,0,0,0; 0,1,0,0] covariance matrix and observation noise covariance matrix R on the prediction results of LSTM-KF algorithm, System noise covariance matrix Q [1,0,0,0; 0,1,0,0; 0,0,1,0; 0,0,0,1] different experimental parameters Q and R are selected for Observation noise covariance [0.3,0; 0, 0.3] the experiment. The RMSE difference between ILSTM and matrix R LSTM-KF is used to measure the impact of Q and R on the Comput Geosci

Fig. 12 Comparison of the trajectory prediction results of the simple LSTM, ILSTM and LSTM-KF trajectory prediction algorithms for hurricane KARL

Fig. 13 Comparison of the trajectory prediction results of the simple LSTM, ILSTM and LSTM-KF trajectory prediction algorithms for hurricane MATTHEW

Fig. 14 Comparison of the trajectory prediction results of the simple LSTM, ILSTM and LSTM-KF trajectory prediction algorithms for hurricane NICOLE Comput Geosci

Fig. 15 Comparison of the trajectory prediction results of the simple LSTM, ILSTM and LSTM-KF trajectory prediction algorithms for hurricane BONNIE

prediction results, as shown in Eq. 23. This neural network Q = [2,0,0,0; 0,2,0,0; 0,0,2,0; 0,0,0,2], the result of structure is set to 256 * 256 * 256, the state transition matrix ΔRMSE changes with R is shown in Fig. 17b. It can be A = [1,0,1,0; 0,1,0,1; 0,0,1,0; 0,0,0,1], the observation seen from Fig. 17 that the observation noise covariance matrix H = [1,0,0,0; 0,1,0,0]. We select hurricane data matrix R will have an impact on the prediction accuracy of from 1895–2015 for training and forecast hurricanes in LSTM-KF algorithm. Figure 17a and b show that ΔRMSE 2016. When the system noise covariance matrix Q are set tend to increase and then decrease rather than increase to [1,0,0,0; 0,1,0,0; 0,0,1,0; 0,0,0,1] and [2,0,0,0; 0,2,0,0; indefinitely as R increases. In Fig. 17a, the maximum 0,0,2,0; 0,0,0,2], the observation noise covariance matrix R, ΔRMSE is 0.0419 when R = R2, and the minimum respectively, using the following parameters are shown in ΔRMSE is -0.0033 when R = R10.InFig.17b, the Table 7. maximum ΔRMSE is 0.0404 when R = R4,andthe minimum ΔRMSE is 0.0077 when R = R . Therefore, epochs 0.1 1 different R will have different influences on the accuracy ΔRMSE = RMSE −RMSE − (23) epochs ILSTM LSTM KF of the predicted trajectory, and if the selected R is not i=1 appropriate, it may also have a negative effect on the Where RMSELST M−KF represents the root mean predicted results. For example, ΔRMSE is negative when square error of LSTM-KF algorithm. RMSEILSTM repre- R = R10 in Fig. 17a. sents the root mean square error of LSTM-KF algorithm. Then, the observation noise R is set as [1, 0; 0, 1], [2, 0; 0, epochs represent the training rounds, and epochs are set to 2], [3, 0; 0, 3], [4, 0; 0, 4], and the system noise covariance 100. matrix Q is set as the parameters shown in Table 8.The When Q = [1,0,0,0; 0,1,0,0; 0,0,1,0; 0,0,0,1], the result result of ΔRMSE changes with system noise covariance of ΔRMSE changes with R is shown in Fig. 17a. When matrix Q asshowninFig.18.

Fig. 16 Comparison of the trajectory prediction results of the simple LSTM, ILSTM and LSTM-KF trajectory prediction algorithms for hurricane ALEX

Comparison 1

Comparison 2 Comput Geosci

Table 7 Parameter Settings of observation noise covariance matrix R Table 8 Parameter Settings of system noise covariance matrix Q

R Value R Value Q Value Q Value

R0.1 [0.1,0; 0, 0.1] R1 [1, 0; 0, 1] Q0.1 [0.1,0; 0, 0.1] Q1 [1,0;0,1] R0.2 [0.2,0; 0, 0.2] R2 [2, 0; 0, 2] Q0.2 [0.2,0; 0, 0.2] Q2 [2,0;0,2] R0.2 [0.3,0; 0, 0.3] R3 [3, 0; 0, 3] Q0.2 [0.3,0; 0, 0.3] Q3 [3,0;0,3] R0.4 [0.4,0; 0, 0.4] R4 [4, 0; 0, 4] Q0.4 [0.4,0; 0, 0.4] Q4 [4,0;0,4] R0.5 [0.5,0; 0, 0.5] R5 [5, 0; 0, 5] Q0.5 [0.5,0; 0, 0.5] Q5 [5,0;0,5] R0.6 [0.6,0; 0, 0.6] R6 [6, 0; 0, 6] Q0.6 [0.6,0; 0, 0.6] Q6 [6,0;0,6] R0.7 [0.7,0; 0, 0.7] R7 [7, 0; 0, 7] Q0.7 [0.7,0; 0, 0.7] Q7 [7,0;0,7] R0.8 [0.8,0; 0, 0.8] R8 [8, 0; 0, 8] Q0.8 [0.8,0; 0, 0.8] Q8 [8,0;0,8] R0.9 [0.9,0; 0, 0.9] R9 [9, 0; 0, 9] Q0.9 [0.9,0; 0, 0.9] Q9 [9,0;0,9] R10 [10, 0; 0, 10] Q10 [10, 0; 0, 10]

When R = [1, 0; 0, 1], the result of ΔRMSE changes results, but the effect of prediction optimization may be in a with Q are shown in Fig. 18a. When R = [2, 0; 0, 2], the small range. result of ΔRMSE changes with Q are shown in Fig. 18b. In order to test the time performance of LSTM-KF When R = [3, 0; 0, 3], the result of ΔRMSE changes algorithm, this experiment tested the time performance by with Q are shown in Fig. 18c, and When R = [4, 0; 0, counting model training time and predicted response time of 4], the result of ΔRMSE changes with Q are shown in different network structures. The hardware platform of this Fig. 18d. As shown in Fig. 18a, b, c, and d, ΔRMSE experiment is CPU Intel(R) Core(TM) i7-9850h, 2.60 Ghz, also approximate a trend of increasing and then decreasing. 16G memory, GPU Quadro RTX 3000. We select hurricane Therefore, different system noise covariance matrix Q will data from 1895 to 2015 for training and statistic the model have an impact on the prediction accuracy of LSTM-KF training time, as shown in Fig. 19. Forecast the hurricane algorithm, and inappropriate values may have a negative trajectory in 2016 and statistic their predicted response impact. For example, in Fig. 18b, c, and d, ΔRMSE is time, as shown in Fig. 20. It can be seen from Fig. 19 negative when Q is compared at the beginning. However, it that the neural network structure is 128*128*256, 256*256 is worth noting that the better effect of Q and R is around *256, 128*256 *512, 256 *256, and the training time of 0.05. As shown in Fig. 17a, the maximum ΔRMSE is the model is 870s, 929s, 913s and 817s, respectively. As 0.0419 when R = R2. As shown in Fig. 18a, the maximum shown in Fig. 20, the structure of the neural network is ΔRMSE is 0.0535 when Q = Q0.6. As shown in Fig. 18b, 128*128*256, 256*256 *256, 128*256 *512, 256 *256, the maximum ΔRMSE is 0.0556 when Q = Q0.9.As and the response time of model is 5s, 6s, 6s and 4s showninFig.18d, the maximum ΔRMSE is 0.0512 respectively. The predicted response time of the whole year when Q = Q5. Selecting the appropriate value of Q and of 2016 has remained at a few seconds. Although LSTM- R will produce better optimization effect on the predicted KF algorithm takes more time in model training, the time

Fig. 17 Result of ΔRMSE changes with observation noise covariance matrix R Comput Geosci

Fig. 18 Result of ΔRMSE changes with system noise covariance matrix Q interval of hurricane trajectory prediction is 6 h, so the prediction process of trajectory is response time. As shown model training time is much lower than the time interval in Fig. 20, the predicted response time of the whole year of between the sampling points. Meanwhile, the LSTM-KF 2016 is very short, only within a few seconds. And the better algorithm only needs to train the trajectory prediction model the hardware platform, the shorter the time. Therefore, before each hurricane arrives and it is not necessary to LSTM-KF algorithm can meet the requirements of real-time train the model for each prediction. The time spent in the hurricane prediction.

Fig. 19 Model training time for different network structures 940 929 920 913 900 880 870 860 840 817 820 800

Model traing time (unit:s) traing Model 780 760 128*128*256 256*256*256 128*256*512 256*256 The structure network of LSTM Comput Geosci

Fig. 20 The predicted response time of different network 7 structures 66 6 5 5 4 time (unit:s) time 4

Predicted response Predicted 0 128*128*256 256*256*256 128*256*512 256*256 The structure network of LSTM

6 Conclusion Declarations

Trajectory prediction has become a research hotspot in Conﬂict of interest The authors declare that they have no conflict of many fields. Hurricanes are serious threats to people’s lives interest. and cause significant economic losses. Effective prediction Open Access This article is licensed under a Creative Commons of the trajectory of a hurricane has good application value. Attribution 4.0 International License, which permits use, sharing, In this paper, from the perspective of machine learning, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the using real Atlantic hurricane data, an LSTM network is source, provide a link to the Creative Commons licence, and indicate applied to hurricane trajectory prediction, and the prediction if changes were made. The images or other third party material in model is trained using historical hurricane data. The main this article are included in the article’s Creative Commons licence, contributions of this paper are as follows: 1. In the stage of unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your data preprocessing, the trajectories are gridded and encoded, intended use is not permitted by statutory regulation or exceeds and a trajectory vector is generated by combining numerical the permitted use, you will need to obtain permission directly from data with categorical data to input into the LSTM. 2. the copyright holder. To view a copy of this licence, visit http:// Based on the simple LSTM trajectory prediction algorithm, creativecommonshorg/licenses/by/4.0/. this paper improves the prediction module and proposes an improved LSTM trajectory prediction algorithm. 3. Combined with a Kalman filter, the predicted coordinates References of the improved LSTM trajectory prediction algorithm are filtered, and the LSTM-KF trajectory prediction algorithm 1. Zheng, Y.: Trajectory data mining: an overview. ACM Trans. is proposed. 4. Real Atlantic hurricane data from 1851 to Intell. Syst. Technol. (TIST) 6(3), 1–41 (2015) 2. Tang, J.: Conflict detection and resolution for civil aviation: 2016 are used in simulation experiments. The prediction a literature survey. IEEE Aerosp. Electron. Syst. Mag. 34(10), results of the LSTM-KF trajectory prediction algorithm 20–35 (2019) are better than those of the improved LSTM algorithm 3. Meuel, T., Prado, G., Seychelles, F., Bessafi, M., Kellay, H.: and the simple LSTM algorithm. The trajectory prediction Hurricane track forecast cones from fluctuations. Sci. Rep. 2, 446 (2012) algorithm proposed in this paper only considers several 4. Lee, R.S., Liu, J.N.: Tropical cyclone identification and tracking factors, such as latitude and longitude, maximum sustained system using integrated neural oscillatory elastic graph matching wind speed and the system state, but its prediction error is and hybrid RBF network track mining techniques. IEEE Trans. still very large. It is hoped that more meteorological factors Neural Netw. 11(3), 680–689 (2000) 5. Rajasekaran, S., Lee, T., Jeng, D.-S.: Tidal level forecasting during can be considered in future studies to establish a more typhoon surge using functional and sequential learning neural complete prediction model. networks. J. Waterway Port Coastal Ocean Eng. 131(6), 321–324 (2005) 6. Rozanova, O.S., Yu, J.-L., Hu, C.-K.: Typhoon eye trajectory Acknowledgements This work was supported by the National Natural based on a mathematical model: comparing with observational Science Foundation of China (62073330), Natural Science Foundation data. Nonlinear Anal.: Real World Appl. 11(3), 1847–1861 of Hunan Province (2019JJ20021), NUDT Scientific Research Project (2010) (ZK18-02-12), Huxiang Young Talents (2018RS3079) and Innovation 7. DeMaria, M., Mainelli, M., Shay, L.K., Knaff, J.A., Kaplan, project for graduate students in Hunan Province (CX20190046). J.: Further improvements to the statistical hurricane intensity Comput Geosci

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